We establish the lower bound , for the large times asymptotic behaviours of the probabilities of return to the origin at even times , for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer , such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to .)
Let F n denote the free group of rank n and d(G) the minimal number of generators of the finitely generated group G. Suppose that R ↪ F m ↠ G and S ↪ F m ↠ G are presentations of G and let and denote the associated relation modules of G. It is well known that even though it is quite possible that . However, to the best of the author’s knowledge no examples have appeared in the literature with the property that . Our purpose here is to exhibit, for each integer k ≥ 1, a group G that has presentations...